An Approximate Lax-Wendroff-Type Procedure for High Order Accurate Schemes for Hyperbolic Conservation Laws

TL;DR

This paper introduces an approximate high-order time-stepping method for hyperbolic conservation laws that simplifies implementation while maintaining accuracy, improving upon previous exact derivative-based techniques.

Contribution

It presents a new finite difference approximation approach for the Lax-Wendroff-type procedure, reducing complexity and computational effort compared to the exact derivative method.

Findings

Achieves arbitrarily high order accuracy in space and time.

Simplifies implementation by avoiding symbolic flux derivative computations.

Demonstrates improved performance over previous methods.

Abstract

A high order time stepping applied to spatial discretizations provided by the method of lines for hyperbolic conservations laws is presented. This procedure is related to the one proposed in Qiu and Shu (SIAM J Sci Comput 24(6):2185-2198, 2003) for numerically solving hyperbolic conservation laws. Both methods are based on the conversion of time derivatives to spatial derivatives through a Lax-Wendroff-type procedure, also known as Cauchy-Kovalevskaya process. The original approach in Qiu and Shu (2003) uses the exact expressions of the fluxes and their derivatives whereas the new procedure computes suitable finite difference approximations of them ensuring arbitrarily high order accuracy both in space and time as the original technique does, with a much simpler implementation and generically better performance, since only flux evaluations are required and no symbolic computations of flux derivatives are needed.